MinT - Best Known (68−52, 68, s)-Nets in Base 64

Best Known (68−52, 68, s)-Nets in Base 64

(68−52, 68, 177)-Net over F₆₄ — Constructive and digital

Digital (16, 68, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 68, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(68−52, 68, 216)-Net in Base 64 — Constructive

(16, 68, 216)-net in base 64, using

9 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
- base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(68−52, 68, 267)-Net over F₆₄ — Digital

Digital (16, 68, 267)-net over F₆₄, using

net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
  - a curve from the manYPoints database [i]

(68−52, 68, 8854)-Net in Base 64 — Upper bound on s

There is no (16, 68, 8855)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 661 587212 484701 421573 218291 928390 897309 696877 819781 969361 468927 369384 958242 944852 641495 916908 361758 534145 068117 296788 094736 > 64⁶⁸ [i]